Ore-degree threshold for the square of a Hamiltonian cycle
Authors: Louis DeBiasio ; Safi Faizullah ^{1,}^{2}; Imdadullah Khan ^{3}
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Louis DeBiasio;Safi Faizullah;Imdadullah Khan
1 Hewlett-Packard
2 Department of Computer Science [Rutgers]
3 Department of Computer Science [Lahore]]
A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.